The well known modulation transfer function (MTF), and phase transfer function (PTF) provide measures of imaging resolution and performance for an imaging system. The MTF and PTF uniquely define the resolution and performance of a particular imaging system to the extent that the imaging system is linear and shift-invariant. The MTF and PTF can be mathematically combined into a single complex-valued quantity called the system transfer function (STF), given by the equation: EQU S=Mexp(i.phi.) (1)
where S is the STF, M is the MTF, and .PHI. is the PTF. The STF, often referred to as the optical transfer function (OTF), may be used interchangeably with OTF.
Assuming linearity and shift-invariance, the STF allows the computation of the image output of the system for any image input, according to the equation: EQU O=F.sup.-1 [S.times.F[I]] (2)
where I is the image input to the system, O is the image output, F is the Fourier Transform operator, and F.sup.-1 is the inverse Fourier Transform operator. Equation (2) implies that if S=1, the input image passes through the imaging system to the output unaltered.
In an optical system, aberrations, improper focus, lens imperfections such as pits or scratches, and foreign material in the optical train, such as dust, can all affect the STF, and hence the image quality. In an imaging system which contains electronic circuitry, the bandwidth and other performance characteristics of the relevant electronic components can also affect the STF.
In an imaging system where images are captured to be electronically processed and classified by a computer, alternately known as an image interpretation system, maintaining a consistent STF becomes much more important. Unless specifically and carefully designed to do so, the image interpretation systems of the prior art will not detect or correct for differences in STF, which can be caused by manufacturing tolerances, environmental variations, or components which have become improperly adjusted or defective.
Therefore, it is one motive of the invention to accurately measure and monitor the STF of imaging systems which are part of an image interpretation system. Without monitoring, the performance of the image interpretation system can vary significantly due to differences in STF. Without accurate measurement, close monitoring of the STF is difficult.
The prior art has found it difficult to achieve accurate measurement of the STF, for several reasons. One significant difficulty is presented by the fact that image interpretation systems are designed to store and process images in a sampled, digital form. Once an image is sampled and becomes discrete, as happens when the image is sensed by a charge coupled device (CCD), or converted to digital form by an analog-to-digital (A/D) converter, Equation (2) no longer completely describes the affect of the system on the image. In particular, the system is no longer shift invariant.
A sampling imaging system is characterized by a sampling function G, defined by ##EQU1## where Y.sub.i denotes the N points at which the image is sampled, x denotes the image domain, and .delta. is the Dirac delta function, which is zero except when its argument vanishes, and which integrates to one over its domain. Using the sampling function G, the system response equation of a sampling image system may be written: EQU O.sub.s =G.times.F.sup.-1 [S.sub.2 .times.F[G.times.F.sup.-1 [S.sub.1 .times.F[I]]]] (4)
where O.sub.s represents the sampled image output, S.sub.1 is a system response function for that part of the system which affects the formation of the image prior to sampling, and S.sub.2 is a second system response function for that part of the system affecting the image after initial sampling. S.sub.2 is relevant to systems in which the image signal is transferred in analog form, even after initial sampling. This is how typical CCD cameras work, for instance.
It may seem at first that there is, in a sampled image system as described by Equation (4), no single function which can be called the STF. The effect of multiplication by the sampling function G in the spatial domain is equivalent to convolution with the Fourier transform of G in the frequency domain. EQU G.times.I=F.sup.-1 [G*I], where (5) EQU G=F[G] (6) EQU I=F[I] (7)
But if the sampling array defined by y.sub.i is periodic, or nearly so over a substantial region, the Fourier transform of G will comprise an array of sharp signal peaks at integer multiples of the sampling frequency. Convolution with this array adds shifted copies of the signal to itself in the the frequency domain, where each copy is shifted by appropriate integer multiple of the sampling frequency.
Thus, if an unsampled image is restricted to contain only frequencies below half the sampling frequency, the corresponding sampled image will be, in the Fourier domain, a copy of the unsampled image, at frequencies below half the sampling frequency. This is the well-known Nyquist theorem, and half the sampling frequency is often called the Nyquist frequency for this reason. From Equation (4), it follows that for input frequencies below the Nyquist frequency, an STF is defined by the equation: EQU S=S.sub.2 .times.S.sub.1 ( 8)
Refer now to FIG. 1 which shows a lens 102, representing part of an imaging system. An image primitive 101, here represented by a narrow slit is imaged by the lens 102 onto the image plane 103. The image plane 103 is defined by a sampling device, here an array of charge-coupled-device (CCD) pixels 104. After sampling, the image is read out of the device and processed to determine its frequency content. FIG. 2 shows part of a representative line from a sampled image of a test pattern like FIG. 1. In this case, the image primitive was a bar pattern, which cycled from white to black and back with a period of about thirty pixels.
One difficulty with this prior art method of measuring the frequency response occurs in the sampling. As described above, if the image primitive contains frequencies above the Nyquist frequency of the sampling array, and those frequencies are passed by the imaging system, they will be added by sampling to the frequency response at lower frequencies, causing the measurement of the frequency response to be inaccurate.
FIG. 3 illustrates this problem, known as aliasing. FIG. 3 is a plot of the amplitude of the Fourier transform of the windowed sampled bar pattern signal, part of which was plotted in FIG. 2. The spectrum comprises a series of signal peaks, reflecting the periodic nature of the signal. The peaks 301, 302, 303, 304, 305, 306, 307 and 308 are the odd harmonics from one to fifteen, respectively. The even harmonics do not appear in this signal because of the symmetry between the light and dark halves of the bar pattern. The peaks beyond the fifteenth harmonic are effectively reflected back from the Nyquist frequency, or aliased. In particular, the peaks 309, 310, 311, 312 and 313 are the image of the odd harmonics from 17 to 25, respectively.
Note that in this signal, the aliased peaks 309, 310, 311, 312 and 313 are distinct in frequency from the unaliased peaks 301, 302, 303, 304, 305, 306, 307 and 308, making accurate measurement of the frequency response possible, both below and above the Nyquist frequency. If the magnification of the imaging system or the period of the bar pattern had been slightly different, however, the aliased peak 309 would directly interfere with the measurement of the peak 307, and so on, rendering inaccurate the measurement of the frequency response below the Nyquist frequency, and destroying any measurement of the frequency response above the Nyquist frequency.
One solution to this problem is to prepare image primitives with strictly band-limited frequency content. For example, sinusoidal image primitives, which provide a signal containing only a single non-zero frequency component, may be employed. Unfortunately, such image primitives are difficult and expensive to produce accurately. Another solution is to use a very fine sampling array, so that the Nyquist frequency lies beyond the cutoff frequency of the imaging system, in order that once again, no signal will pass beyond the Nyquist frequency. However, this solution will often require very high resolution cameras, which are expensive, and slow to read out.